Comparison and Positive Solutions for Problems with the (p, q)-Laplacian and a Convection Term

2014 ◽  
Vol 57 (3) ◽  
pp. 687-698 ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olímpio H. Miyagaki ◽  
Dumitru Motreanu
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Comparison Principle ◽  
Elliptic Problem ◽  
Principal Part ◽  
Convection Term ◽  
Gradient Bounds ◽  
Linear Elliptic Problem

AbstractThe aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p, q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.

On the Structure of Positive Solutions for the Shigesada–Kawasaki–Teramoto Model with Large Interspecific Competition Rate

2020 ◽  
Vol 30 (01) ◽  
pp. 2050001
Author(s):  
Yukio Kan-on
Keyword(s):  
Positive Solution ◽  
Interspecific Competition ◽  
Positive Solutions ◽  
Comparison Principle ◽  
Nonlinear Diffusion ◽  
Bifurcation Theory ◽  
Diffusion System ◽  
Bifurcation Structure ◽  
Diffusion Term ◽  
Competition Diffusion

In this paper, we treat the competition-diffusion system with nonlinear diffusion term, which was proposed by Shigesada et al. [1979], and discuss the bifurcation structure of positive solution for the system when the interspecific competition rate is sufficiently large. To do this, we derive two kinds of limiting systems as the interspecific competition rate tends to infinity, and study the bifurcation structure of positive solution for each limiting system by employing the comparison principle and the bifurcation theory.

scholarly journals A Sub-Supersolution Approach for Robin Boundary Value Problems with Full Gradient Dependence

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 658 ◽  
Author(s):  
Dumitru Motreanu ◽  
Angela Sciammetta ◽  
Elisabetta Tornatore
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Robin Boundary Condition ◽  
Convection Term ◽  
Nonlinear Elliptic Problem ◽  
Existence Of A Solution ◽  
Nonlinear Elliptic ◽  
Robin Boundary

The paper investigates a nonlinear elliptic problem with a Robin boundary condition, which exhibits a convection term with full dependence on the solution and its gradient. A sub- supersolution approach is developed for this type of problems. The main result establishes the existence of a solution enclosed in the ordered interval formed by a sub-supersolution. The result is applied to find positive solutions.

Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition

2016 ◽  
Vol 19 (01) ◽  
pp. 1550090 ◽  
Author(s):  
Shouchuan Hu ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Boundary Condition ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Type Result ◽  
Unbounded Potential ◽  
Concave Boundary ◽  
Negative Laplacian ◽  
Superlinear Reaction

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical “convex–concave” problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter [Formula: see text] varies. We also show the existence of a smallest positive solution [Formula: see text] and investigate the properties of the map [Formula: see text]. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.

Existence of Multiple Solutions for a p-Laplacian System in ℝN with Sign-changing Weight Functions

2016 ◽  
Vol 59 (2) ◽  
pp. 417-434 ◽  
Author(s):  
Hongxue Song ◽  
Caisheng Chen ◽  
Qinglun Yan
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Multiple Solutions ◽  
Continuous Functions ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Image Position ◽  
The Nehari Manifold ◽  
Linear Elliptic Problem

AbstractIn this paper, we consider the quasi-linear elliptic problemwhere and the weight H(x); h1(x); h2(x) are continuous functions that change sign in ℝN. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

Positive Solutions of a Nonlocal and Nonvariational Elliptic Problem

2021 ◽  
Vol 41 (5) ◽  
pp. 1764-1776
Author(s):  
Lingjun Liu ◽  
Feilin Shi
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

Sign in / Sign up

Export Citation Format

Share Document